Embeddings of cubic Halin graphs: Genus distributions

Abstract

We derive an O ( n 2 )-time algorithm for calculating the genus distribution of a given 3-regular Halin graph G ; that is, we calculate the sequence of numbers g 0 ( G ), g 1 ( G ), g 2 ( G ), … on the respective orientable surfaces S 0 , S 1 , S 2 , …. Key topological features are a quadrangular decomposition of plane Halin graphs and a new recombinant-strands reassembly process that fits pieces together three-at-a-vertex. Key algorithmic features are reassembly along a post-order traversal , with just-in-time dynamic assignment of roots for quadrangular pieces encountered along the tour.